MATHEMATICAL MODELS - Vol. I - Controllability, Observability, and Stability of Mathematical Models - Abderrahman Iggidr

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1 CONTROLLABILITY, OBSERVABILITY, AND STABILITY OF MATHEMATICAL MODELS Abderrahma Iggidr INRIA (Ur Lorraie) ad, Uiversity of Metz, Frace Keywords: accessibility, asymptotic stability, attractivity, chemostat, closed-loop, cotrol-lability, differetial equatio, fiite dimesioal systems, iput, Lie algebra, Lotka-Volterra systems, Lyapuov fuctios, oliear systems, observability, observer, output, stabilizatio, state. Cotets 1. Itroductio 2. Cotrollability 2.1. What is Cotrollability? 2.2. Cotrollability of Liear Systems 2.3. Accessibility Criteria for Noliear Systems 3. Stability 3.1. Geeral Defiitios 3.2. Stability of Liear Systems 3.3.Liearizatio ad Stability of Noliear Systems 3.4. Lyapuov Fuctios 3.5. Limit Cycle 3.6. Stabilizatio Sufficiet Stabilizability Coditios 4. Observability 4.1. Observability of Liear Systems 4.2. Observability of Noliear Systems 4.3. Examples from Life Support Systems 5. Observers 5.1. Observers for Liear Systems 5.2. Some Noliear Observers System with o Iput Affie Cotrol Systems Observers for a Class of No-affie Cotrol Systems Asymptotic Observers Ackowledgemets Glossary Bibliography Biographical Sketch Summary This chapter presets a overview of three fudametal cocepts i Mathematical System Theory: cotrollability, stability ad observability. These properties play a promiet role i the study of mathematical models ad i the uderstadig of their behavior. They costitute the mai research subject i Cotrol Theory. Historically the

2 tools ad techiques of Automatic Cotrol have bee developed for artificial egieerig systems but owadays they are more ad more applied to atural systems. The mai objective of this chapter is to show how these tools ca be helpful to model ad to cotrol a wide variety of atural systems. 1. Itroductio The mai goal of this chapter is to develop i some details some otios of Cotrol Theory itroduced i Basic priciples of mathematical modelig. It cocers more specifically three structural properties of cotrol systems: cotrollability, stability ad observability. Based o the refereces listed i the ed of this chapter, we give a survey of these three properties with applicatios to various LSS examples. By a cotrol system we mea dyamical system evolvig i some state space ad that ca be cotrolled by the user. More precisely we are iterested i the study of systems that ca be modeled by differetial (respectively differece) equatios of the form dx x () t = = X ( x() t, u() t ), y() t = h( x() t ), where the variable t represets the time, the vector x(t) is the state of the system at time t, the vector u(t) is the iput or the cotrol, i.e., the actio of the user or of the eviromet ad the vector y() t is the output of the system that is, the available iformatio that ca be measured or observed by the user. The dyamics fuctio X idicates how the system chages over time. We shall address the followig problems: Whe the whole state x(t) is ot available for measuremet, how is it possible to use the iformatio provided by y( t ) together with the dyamics (1) i order to get a good estimatio of the real state of the system? This turs out to be a observability problem. How to use the cotrol u(t) i order to meet some specified eeds? This is the problem of cotrollability ad stabilizability. (1) To illustrate these various cocepts, we explai them through a simple LSS system: a epidemic model for the trasmissio of a ifectious disease. A homogeeous populatio is divided ito four classes S, E, I ad T accordig to the health of its idividuals. Let S(t) deote the umber of idividuals who are susceptible to the disease, i.e., who are ot yet ifected at time t. E(t) deotes the umber of members at time t who are exposed but ot yet ifected. I(t) deotes the umber of ifected idividuals, that is,

3 who are ifectious ad able to spread the disease by cotact with idividuals who are susceptible. T(t) is the umber at time t of treated idividuals. The total populatio size is deoted N = S + E + I + T. The dyamic of the disease ca be described by the followig differetial system: ds SI = bn μs β, N de SI = β ( μ + ε) E, N di = εe ( r + d + μ) I, dt = ri μt, dn = ( b μ ) N di, where the parameter b is the rate for atural birth ad μ that of atural death. The parameter β is the trasmissio rate, d is the rate for disease-related death, ε is the rate at which the exposed idividuals become ifective ad r is the per-capita treatmet rate. The parameters b, μ, β, d ad ε are assumed to be costat. The per-capita treatmet rate may vary with time. (2) Fig 1: This diagram illustrates the dyamical trasfer of the populatio. The system (2) ca be writte by usig the fractios s = S N, e = E N, i = I ad τ = T N of the classes S, E, I ad T i the populatio. These fractios satisfy the system of differetial equatios:

4 ds = b bs ( β d) si, de βsi ( b ε) e dei, = + + di εe r d b i di = dτ = ri bτ + diτ. ( ) 2, (3) Sice s+ e+ i+ τ = 1, it is sufficiet to cosider the followig system ds = b bs ( β d) si, de = βsi ( b + ε) e + dei, di 2 = εe ( r+ d + b) i+ di. If we suppose that the proportio of ifected idividuals ca be measured ad that we ca cotrol the populatio by choosig the treatmet rate the the above system ca be see as a cotrol system of the form (1) with: the state of the system is x t s t, e t, i t u t = r t ad the measurable output is () = ( () () ()), the cotrol is ( ) ( ) () = i() t. The state space is y t { x 3 : 0 s 1, 0 e 1, 0 i 1, s e i 1} Ω= + +. The aim of Cotrol Theory is to give aswers to the followig questios: 1. Give two cofiguratios x 1 ad x 2 of the system (3), is it possible to steer the x system from 1 to x 2 by choosig a appropriate treatmet strategy r(t)? For a x give iitial state 0 = ( s0, e0, i0) ad a give time T, what are all the possible states that ca be reached i time T by usig all the possible cotrol fuctios r(t)? 2. Does the system (3) possess equilibrium poits? Are they stable or ustable? How ca the cotrol be chose i order to make a give equilibrium (for istace, the disease-free equilibrium) globally asymptotically stable? 3. Sice oly some variables ca be measured (for istace, the proportio i(t) of ifected idividuals i the above epidemic example), is it possible to use them together with the dyamics (3) of the system i order to estimate the o measurable variables s(t) ad e(t) ad how ca this be doe? (4) Questios 1 ivolve the cotrollability of cotrol systems. We shall give i Sectio 2 a geeral defiitio of this otio as well as some simple useful criteria that allow us to

5 study the cotrollability of some oliear systems. Questios 2 ivoke the stability ad the stabilizatio of oliear system, this will be the subject of Sectio 3 where some classical stability results are exposed. Questios 3 are coected to the observability ad the costructio of observers for dyamical systems. These problems will be addressed i Sectio 4 ad Sectio 5. I each sectio we shall apply the differet tools to various models such as predatorprey systems, fisheries ad bioreactors. Cocerig the epidemic model (2-4), the divers problems metioed above are still uder ivestigatio. We ca give here just some partial aswers. It ca be show that if the per-capita treatmet rate satisfies βε r r0 = d b ε + b the the system has a uique equilibrium poit i the domai Ω. This equilibrium is the disease free equilibrium s = 1, e = i = τ = 0 ad it is globally asymptotically stable, that is, if ( s( 0, ) e( 0, ) i( 0) ) Ω is ay iitial coditio the the solutio of the system (4) startig from this state will coverge to the disease free equilibrium, i.e., s() t 1, e() t 0adi() t 0ast. Therefore, if the treatmet satisfies the coditio (5) the the disease will die out. If r < r0 the the disease free equilibrium becomes ustable ad i this case, there is aother equilibrium which belogs to the iterior of the domai Ω. This equilibrium is the edemic equilibrium ad it is globally asymptotically stable withi the iterior of Ω provided that r < r0. The disease will be edemic i this case. 2. Cotrollability I this sectio, we preset a elemetary overview of a importat property of a system, amely that of cotrollability. This cocept has bee briefly itroduced ad explaied i Basic priciples of mathematical modelig. Here, we deal with the problem of testig cotrollability of systems of the type () = ( (), ()) x t X x t u t. (6) The state vector x(t) belogs to the state space M which will always be here ope coected subset of t u t (5) or a. The cotrol fuctio ( ) are defied o [ 0, ) ad take values i a coected subset U of. These cotrol fuctios are assumed to belog to a admissible cotrol set U ad. This admissible cotrol set is geerally specified by the problem cosidered. Here it is assumed that U ad cotais the set of piecewise costat fuctios as a dese subset, i.e., ay admissible cotrol fuctio ca be approximated by piecewise costat fuctios. The dyamics fuctio X : M x, u. For each cotrol value u U, is assumed to be aalytic o ( ) m

6 we deote by all x M. u u X the vector field ( vector fuctio ) defied by X ( x) X ( x, u) = for A real-valued fuctio f defied o some ope set O is said to be aalytic if for each x0 O, there exists a positive real umber r > 0 such that f(x) is the sum of a power series for all x with x x0 < r: 0 0. = 0 ( ) ( ) f x = a x x for all x satisfyig x x < r For a aalytic fuctio, the coefficiets a ca be computed as k a = ( ) ( a) f! I a similar way a fuctio f : V of several variables is said to be aalytic if it is locally give by power series. A vector fuctio X : V k (,..., ) X1 x1 xk x = ( x1,..., xk ) X ( x) = X x1 xk is aalytic if all its compoets i (,..., ) X are aalytic. u. For each x M ad each cotrol fuctio u (.) U ad, we deote by Xt x the u(). u solutio of (6) satisfyig X0 ( x) = x. We assume moreover that (). Xt ( x ) is defied for all t [ 0, ). For istace, if X is a liear vector fuctio, that is, X ( x, u) = Ax + Bu, with A ad B beig matrices with appropriate dimesios, the u(). ta t ( ) ( t s) A X x = e x+ e Bu s ds t 0 ( ) ( ) ( ) TO ACCESS ALL THE 70 PAGES OF THIS CHAPTER, Visit:

7 Bibliography Bacciotti, A. (1992). Local stabilizability of oliear cotrol systems, 202 p. Series o Advaces i Mathematics for Applied Scieces. 8. Sigapore, World Scietific. [A well writte ad easy-to-read itroductio to the stabilizatio of cotrol systems]. Basti G. ad Dochai D. (1990). O-lie Estimatio ad Adaptive Cotrol of Bioreactors, 379 p. Elsevier Publisher. [Bioreactor models have specific structures which are thoroughly cosidered i this book] Berard O., Sallet G. ad Sciadra A. (1998). Noliear Observer for a Class of Biological Systems: Applicatio to Validatio of a Phytoplaktoic Growth Model. IEEE Tras. Automat. Cotrol, 43(8), pp [This paper presets a applicatio of the observers theory to a biological system i order to estimate some state variables that ca ot be measured]. Borard G., Celle-Couee F. ad Gilles G. (1993). Observabilité et observateurs. Systèmes o liéaires, 1. modélisatio estimatio (ed. A.J. Fossard ad D. Normad-Cyrot), pp Masso. [This paper (i Frech) is a readable itroductio to the observability ad observers theory]. Brockett R.W. (1983). Asymptotic stability ad feedback stabilizatio. i Differetial Geometric Cotrol Theory (Ed. Brockett R.W., Millma R.S. ad Sussma H.J.), pp Birkhäuser. [This semial article gives some ecessary coditios for the existece of a stabilizig feedback for a oliear system]. Gauthier J.-P., Hammouri H. ad Othma S. (1992). A simple observer for oliear systems, applicatios to bioreactors. IEEE Tras. Automat. Cotrol, 37(6), pp [This article provides a expoetial observer for the chemostat example with aother absorbig respose]. Gauthier, J.-P., ad Kupka, I. (2001). Determiistic Observatio Theory ad Applicatios, Cambridge Uiversity Press. [This self cotaied book presets the theoretical foudatios of observability. It requires a good backgroud i differetial geometry]. Jurdjevic, V. (1997). Geometric cotrol theory, 492 p. Cambridge Studies i Advaced Mathematics. 52. Cambridge Uiversity Press. [This book presets the essetial aspects of oliear cotrol theory icludig optimal cotrol]. Nijmeijer H. ad Va der Schaft A.J. (1990). Noliear Dyamical Cotrol Systems, 467 pp. New York, Spriger-Verlag. [This book presets may techiques for the study of iput-output oliear systems]. Touzeau S. ad Gouzé J.-L. (1998). O the stock-recruitmet relatioships i fish populatio models. Evirometal modelig ad assessmet, 3: [This article gives a stage-structured model of a fish populatio]. Biographical Sketch Abderrahma Iggidr, was bor i Rabat (Morocco). He received the D.E.A i 1989 i applied mathematics ad the Ph.D. degree i 1992, i mathematics ad automatic cotrol, from the Uiversity of Metz (Frace). Sice 1993, he has bee researcher at INRIA (The Frech Natioal Istitute for Research i Computer Sciece ad Cotrol). His research iterests are the field of oliear systems, cotrollability, observability, stabilizatio, observers, ad stabilizatio by estimated state feedback from observers. His research iterests also iclude the applicatio of cotrol theory to the modelig ad the regulatio of biological systems.